x=seq(-3,3,by=0.1)
plot(x,pnorm(x),type="l")
x=-3:0.1:3;
plot(x,normcdf(x))
set.seed(12) # set seed
S=10
runif(S)
rnorm(S)
rt(S,4)
rng default; % set seed
S=10;
rand(S,1)
randn(S,1)
trnd(4,S,1)
yield=c(5.00, 5.69, 6.09, 6.38, 6.61,
6.79, 6.94, 7.07, 7.19, 7.30) # yield curve
r=0.07 # initial yield rate
Par=10 # par value
coupon=r*Par # coupon payments
cc=1:10*0+coupon # vector of cash flows
cc[10]=cc[10]+Par # add par to cash flows
P=sum(cc/((1+yield/100)^(1:length(yield)))) # calculate price
print(P)
yield = [5.00 5.69 6.09 6.38 6.61...
6.79 6.94 7.07 7.19 7.30]; % yield curve
T = length(yield);
r=0.07; % initial yield rate
Par=10; % par value
coupon=r*Par; % coupon payments
cc=zeros(1,T)+coupon; % vector of cash flows
cc(10)=cc(10)+Par; % add par to cash flows
P=sum(cc./((1+yield./100).^(1:T))) % calculate price
set.seed(12) # set seed
sigma = 1.5 # daily yield volatiltiy
S = 8 # number of simulations
r = rnorm(S,0,sigma) # generate random numbers
ysim = matrix(nrow=length(yield),ncol=S)
for (i in 1:S) ysim[,i]=yield+r[i]
matplot(ysim,type='l')
randn('state',123); % set the seed
sigma = 1.5; % daily yield volatility
S = 8; % number of simulations
r = randn(1,S)*sigma; % generate random numbers
ysim=nan(T,S);
for i=1:S
ysim(:,i)=yield+r(i);
end
ysim=repmat(yield',1,S)+repmat(r,T,1);
plot(ysim)
SP = vector(length=S)
for (i in 1:S){ # S simulations
SP[i] = sum(cc/((1+ysim[,i]/100)^(length(yield))))
}
SP = SP-(mean(SP) - P) # correct for mean
par(mfrow=c(1,2), pty="s")
barplot(SP)
hist(SP,probability=TRUE)
x=seq(6,16,length=100)
lines(x, dnorm(x, mean = mean(SP), sd = sd(SP)))
SP = nan(S,1);
for s = 1:S % S simulations
SP(s) = sum(cc./(1+ysim(:,s)'./100).^((1:T)));
end
SP = SP-(mean(SP) - P); % correct for mean
bar(SP)
P0 = 50 # initial spot price
sigma = 0.2 # annual volatility
r = 0.05 # annual interest
TT = 0.5 # time to expiration
X = 40 # strike price
f = bs(X,P0,r,sigma,length(yield)) # analytical call price
## this calculation uses the Black-Scholes pricing function (Listing 6.1/6.2)
print(f)
P0 = 50; % initial spot price
sigma = 0.2; % annual volatility
r = 0.05; % annual interest
T = 0.5; % time to expiration
X = 40; % strike price
f = bs(X,P0,r,sigma,T) % analytical call price
%% this calculation uses the Black-Scholes pricing function (Listing 6.1/6.2)
set.seed(12) # set seed
S = 1e6 # number of simulations
F = P0*exp(r*length(yield)) # futures price
ysim = rnorm(S,-0.5*sigma^2*length(yield),sigma*sqrt(length(yield))) # sim returns, lognorm corrected
F=F*exp(ysim) # sim futures price
SP = F-X # payoff
SP[SP<0] = 0 # set negative outcomes to zero
fsim = SP*exp(-r*length(yield)) # discount
call_sim = mean(fsim) # simulated price
print(call_sim)
randn('state',0); % set seed
S = 1e6; % number of simulations
F = P0*exp(r*T); % futures price
ysim=randn(S,1)*sigma*sqrt(T)-0.5*T*sigma^2; % sim returns, lognorm corrected
F=F*exp(ysim); % sim futures price
SP = F-X; % payoff
SP(find(SP < 0)) = 0; % set negative outcomes to zero
fsim =SP*exp(-r*T) ; % discount
mean(fsim) % simulated price
par(mfrow=c(1,2), pty="s")
hist(F,probability=TRUE,ylim=c(0,0.06))
x=seq(min(F),max(F),length=100)
lines(x, dnorm(x, mean = mean(F), sd = sd(SP)))
hist(fsim,nclass=100,probability=TRUE)
subplot(1,2,1)
histfit(F);
subplot(1,2,2)
hist(fsim,100);
set.seed(1) # set seed
S = 1e7 # number of simulations
s2 = 0.01^2 # daily variance
p = 0.01 # probability
r = 0.05 # annual riskfree rate
P = 100 # price today
ysim = rnorm(S,r/365-0.5*s2,sqrt(s2)) # sim returns
Psim = P*exp(ysim) # sim future prices
q = sort(Psim-P) # simulated P/L
VaR1 = -q[p*S]
print(VaR1)
randn('state',0); % set seed
S = 1e7; % number of simulations
s2 = 0.01^2; % daily variance
p = 0.01; % probability
r = 0.05; % annual riskfree rate
P = 100; % price today
ysim = randn(S,1)*sqrt(s2)+r/365-0.5*s2; % sim returns
Psim = P*exp(ysim); % sim future prices
q = sort(Psim-P); % simulated P/L
VaR1 = -q(S*p)
TT = 0.25; # time to expiration
X = 100; # strike price
sigma = sqrt(s2*250); # annual volatility
f = bs(X,P,r,sigma,length(yield)) # analytical call price
fsim = bs(X,Psim,r,sigma,length(yield)-(1/365)) # sim option prices
q = sort(fsim$Call-f$Call) # simulated P/L
VaR2 = -q[p*S]
print(VaR2)
T = 0.25; % time to expiration
X = 100; % strike price
sigma = sqrt(s2*250); % annual volatility
f = bs(X,P,r,sigma,T); % analytical call price
fsim=bs(X,Psim,r,sigma,T-(1/365)); % sim option prices
q = sort(fsim.Call-f.Call); % simulated P/L
VaR2 = -q(p*S)
X1 = 100
X2 = 110
f1 = bs(X1,P,r,sigma,TT)
f2 = bs(X2,P,r,sigma,TT)
f2sim = bs(X2,Psim,r,sigma,TT-(1/365))
f1sim = bs(X1,Psim,r,sigma,TT-(1/365))
q = sort(f1sim$Call+f2sim$Put+Psim-f1$Call-f2$Put-P);
VaR3 = -q[p*S]
print(VaR3)
X1 = 100;
X2 = 110;
f1 = bs(X1,P,r,sigma,T);
f2 = bs(X2,P,r,sigma,T);
f1sim=bs(X1,Psim,r,sigma,T-(1/365));
f2sim=bs(X2,Psim,r,sigma,T-(1/365));
q = sort(f1sim.Call+f2sim.Put+Psim-f1.Call-f2.Put-P);
VaR3 = -q(p*S)
library (MASS)
set.seed(12) # set seed
mu = c(r/365,r/365) # return mean
Sigma = matrix(c(0.01,0.0005,0.0005,0.02),ncol=2) # covariance matrix
y = mvrnorm(S,mu,Sigma) # simulated returns
randn('state',12) % set seed
mu = [r/365 r/365]'; % return mean
Sigma=[0.01 0.0005; 0.0005 0.02]; % covariance matrix
y = mvnrnd(mu,Sigma,S); % simulated returns
K=2
P = c(100,50) # prices
x = c(1,1) # number of assets
Port = P %*% x # portfolio at t
Psim = matrix(t(matrix(P,K,S)),ncol=K)*exp(y) # simulated prices
PortSim = Psim %*% x # simulated portfolio value
q = sort(PortSim-Port[1,1]) # simulated P/L
VaR4 = -q[S*p]
print(VaR4)
K = 2;
P = [100 50]'; % prices
x = [1 1]'; % number of assets
Port = P'*x; % portfolio at t
Psim = repmat(P,1,S)' .*exp(y); % simulated prices
PortSim=Psim * x; % simulated portfolio value
q = sort(PortSim-Port); % simulated P/L
VaR4 = -q(S*p)
f = bs(P[2],P[2],r,sigma,TT)
fsim = bs(P[2],Psim[,2],r,sigma,TT-(1/365))
q = sort(fsim$Call+Psim[,1]-f$Call-P[1]);
VaR5 = -q[p*S]
print(VaR5)
f = bs(P(2),P(2),r,sigma,T);
fsim=bs(P(2),Psim(:,2),r,sigma,T-(1/365));
q = sort(fsim.Call+Psim(:,1)-f.Call-P(1));
VaR5 = -q(p*S)